Heat equation and stable minimal Morse functions … · Heat equation and stable minimal Morse functions on real ... Operador de Laplace-Beltrami, ... proof of the analogous result

Revista Colombiana de MatematicasVolumen 51(2017)1, paginas 71-82

Heat equation and stable minimal

Morse functions on real and complex

projective spaces

Ecuacion del calor y funciones de Morse minimales y estables enespacios proyectivos reales y complejos

Sebastian Munoz Munoz1,B,Alexander Quintero Velez2

1Universidad del Valle, Cali, Colombia

2Universidad Nacional de Colombia Sede Medellın, Medellın,Colombia

Abstract. Following similar results in [7] for flat tori and round spheres, inthis paper is presented a proof of the fact that, for “arbitrary” initial con-ditions f0, the solution ft at time t of the heat equation on real or complexprojective spaces eventually becomes (and remains) a minimal Morse function.Furthemore, it is shown that the solution becomes stable.

Key words and phrases. Heat equation, Laplace-Beltrami operator, MinimalMorse function, Fubini-Study metric, Stable function.

2010 Mathematics Subject Classification. 37B30, 53C44, 58J35.

Resumen. Siguiendo resultados similares en [7] para toros planos y esferasredondas, en este artıculo se presenta una demostracion del hecho de que,para condiciones iniciales “arbitrarias” f0, la solucion ft en el tiempo t de laecuacion del calor en espacios proyectivos reales y complejos eventualmente seconvierte en (y permanece siendo) una funcion de Morse minimal con valorescrıticos distintos. Ademas, se muestra que la solucion se vuelve una funcionestable.

Palabras y frases clave. Ecuacion del calor, Operador de Laplace-Beltrami,Funcion de Morse minimal, Metrica de Fubini-Study, Funcion estable.

71

72 SEBASTIAN MUNOZ MUNOZ & ALEXANDER QUINTERO VELEZ

1. Introduction

The heat equation, which is the mathematical model for heat flow, and in itssimplest form is written as

∂f

∂t= ∆f,

where ∆ is the Laplace-Beltrami operator, is among the most relevant partialdifferential equations in mathematics and physics, and it has truly remarkableproperties. For instance, if an “arbitrary” initial condition f0 is subjected tothis equation, then the solution ft, for any instant of time t > 0, will be aC∞ function. In the light of such facts, the heat equation presents itself as afundamental smoothing process.

In a recent paper [7], a result in this direction was established, furtherevidencing important smoothing properties of this equation. Given a smoothfunction f defined on a manifold M , an indicator of regularity for f is whetherall of its critical points are non-degenerate, which is to say that f is a Morsefunction. The authors showed (see Theorem 2.1 and Theorem 3.1 of that paper)that, at least in the case of n-dimensional round spheres or flat tori, “arbitrary”smooth initial conditions f0 are eventually transformed by the heat equationinto minimal Morse functions; Morse functions that have the smallest possiblenumber of critical points in the manifold.

Investigating the question of whether this is a more general phenomenonpresenting itself in other compact Riemannian manifolds, this paper presentsproof of the analogous result for real and complex projective spaces of arbitrarydimension. We will further observe that in these spaces, the heat process endowsthe solution ft with a fundamental property called stability, which roughlyspeaking means that functions close to ft will be “identical” to ft modulo asuitable change of coordinates.1 This is made precise by the following definition.

Definition 1.1. Let M be a compact smooth manifold, and let f ∈ C∞(M).f is said to be stable if there exists a neighborhood Wf of f in the WhitneyC∞ topology such that for each f ′ ∈Wf there exist diffeomorphisms g, h suchthat the following diagram commutes:

M R

M R

f

g h

f ′

The corollary to the following fundamental theorem (see [4, pp. 79-80]) gives asimple characterization of stable functions which will be key to what follows.

1Close in the standard sense of the Whitney C∞ topology, which measures functions bytheir size and that of their partial derivatives of all orders.

Volumen 51, Numero 1, Ano 2017

HEAT EQUATION AND MINIMAL MORSE FUNCTIONS ON PROJECTIVE SPACES 73

Theorem 1.2 (Stability Theorem). Let M be a compact smooth manifold, andf ∈ C∞(M). Then f is a Morse function with distinct critical values if andonly if it is stable.

Corollary 1.3. If M is a smooth compact manifold, and f is a Morse functionwith distinct critical values, then there exists a neighborhood of f in the C∞

topology such that g is a Morse function with distinct critical values and thesame number of critical points as f for all g in said neighborhood. In particular,since M is compact (so that the C∞ topology has the union of all Cr topologiesas a basis, each a Banach space [5]), there exist r and ε > 0 such that g isa Morse function with distinct critical values and the same number of criticalpoints as f whenever ‖f − g‖r < ε , with ‖ · ‖r being a fixed norm for the Cr

topology.

Now, our precise formulation of an “arbitrary” smooth initial condition f0

will be that it belongs to a fixed open and dense set S in the C∞ topology. Wecan now state the two main results.

Theorem 1.4. There exists a set S ⊂ C∞(RPn), that is dense and open inthe C∞ topology, such that for any initial condition f = f0 ∈ S, if ft is thecorresponding solution to the heat equation on RPn at time t, then there existsT > 0 such that for t ≥ T , ft is a stable minimal Morse function on RPn.

Theorem 1.5. There exists a set S ⊂ C∞(CPn), that is dense and openin the C∞ topology, such that for any initial condition f0 ∈ S, if ft is thecorresponding solution to the heat equation on CPn at time t, then there existsT > 0 such that for t ≥ T , ft is a stable minimal Morse function on CPn.

The basic strategy for the proof on both spaces may be sketched as follows:On compact Riemannian manifolds, as is well known, the solution has the form

ft = h0 + h1e−λ1t + h2e

−λ2t + · · · ,

where 0 = λ0 < λ1 < λ2 < · · · are the eigenvalues for the Laplace-Beltrami op-erator, and the hi are the projections of f0 onto the corresponding eigenspaces.The overall idea is to exploit the exponentially decaying terms of the solutionto approximate ft by the first two terms of the sum for large times. Our setS will then consist of those functions for which said first two terms add upto a stable, minimal Morse function. Since stability is by definition a propertyunchanged by small perturbations, it is apparent that the desired result followsprovided that

(i) S is dense and open, and

(ii) we successfully bound the remaining terms to make the approximationwork.

Revista Colombiana de Matematicas


In section 2, simple conditions for the main result to hold in a manifoldwill be established, as well as a technical statement that will be needed later toverify the above-stated conditions on the two spaces in question. On the otherhand, sections 3 and 4 are, respectively, dedicated to the proofs of Theorems1.4 and 1.5.

2. Basic results

The following lemma gives two concrete sufficient conditions for items (i) and(ii) of the introduction to be true, and thereafter makes rigorous the previouslysketched strategy to prove (using some simple estimates and the profound char-acterization of stability given by Theorem 1.2) that the desired theorem willfollow on compact Riemannian manifolds in which the eigenvalues grow at leastlinearly as soon as we have these two sufficient conditions.

Lemma 2.1. Let M be a compact, connected Riemannian manifold, let 0 =λ0 < λ1 < λ2 < · · · be the distinct eigenvalues for the Laplace-Beltrami opera-tor and let B = {ϕi | i = 1, . . . , d1} be any basis (not necessarily orthonormal)for the λ1-eigenspace. Suppose the eigenvalues grow at least linearly,2 that the0-eigenspace is trivial,3 and the following two conditions hold:

(C1) The set B of d1-tuples (c1, . . . , cd1) such that∑i ciϕi is a minimal Morse

function on M with distinct critical values is an open dense subset of Rd1 .

(C2) For each f ∈ C∞(M), there exist N , C such that the projection hj =πj(f) of f onto the jth eigenspace satisfies

‖hj‖r ≤ C(1 + jN ).

Then there exists a set S ⊂ C∞(M), that is dense and open in the C∞ topology,such that for any initial condition f0 ∈ S, if ft is the corresponding solution tothe heat equation on M at time t, then there exists T > 0 such that for t ≥ T ,ft is a minimal Morse function with distinct critical values on M .

Proof. Let S be the set of functions f whose projection π1(f) onto the λ1-eigenspace is a minimal Morse function with distinct critical values. Let f ∈ S.By compactness, ‖·‖L2(M) = O(‖·‖0), so functions in a sufficiently small neigh-borhood U of f in the C0 topology (which is contained in the C∞ topology)will have its Fourier coefficients (with respect to any fixed orthonormal basis)as close as desired to those of f . Observing that the coefficients with respectto B are continuous functions of the Fourier coefficients for the λ1-eigenspace,condition (C1) implies that if U is small enough, U ⊂ S, hence S is open. Now

2By this we mean that there exists r > 0 such that λj > rj.3This means that it consists only of constant functions. This is known to hold on real

and complex projective spaces, basically because it holds on the sphere. This condition is notessential at all; see the next footnote.



let f ∈ C∞(M), and let g be obtained from f such that πi(g) = πi(f) (i 6= 1)and π1(g) comes from slightly modifying the coefficients of π1(f) with respectto B so that π1(g) is a minimal Morse function with distinct critical values (thisis again possible by condition (C1)). By compactness of M , if the modificationis slight enough, g will be as close as desired to f in any Cr (and so in the C∞)topology. So S is dense. Now we check that if f ∈ S and f = h0 +h1 + · · · withhi = πi(f), then ft = h0 +e−λ1th1 + · · · is Morse minimal with distinct criticalvalues for large enough t. Since h0 is constant it suffices to show the same for(ft − h0)eλ1t = h1 + e(λ1−λ2)th2 + · · · , and by Corollary 1.3 it is enough toprove that for each r, ‖(ft − h0)eλ1t − h1‖r → 0 as t→∞. One has

‖(ft − h0)eλ1t − h1‖r =‖h2e(λ1−λ2)t + · · · ‖r

=e(λ1−λ2)t

∥∥∥∥∥∥∑j≥2

e(λ2−λj)thj

∥∥∥∥∥∥r

≤e(λ1−λ2)t∑j≥2

e(λ2−λj)t‖hj‖r

≤e(λ1−λ2)tC∑j≥2

(1 + jN )e(λ2−λj)t.

Because the λj grow at least linearly, the series on the right is clearly convergentand a decreasing function of t, and the first factor tends to zero as t→∞, sothis completes the proof.4 �X

Since the real and complex projective spaces, besides conditions (C1) and(C2), are easily seen to satisfy the hypotheses of the above lemma,5 the remain-ing major part of this paper is dedicated to proving that the aforementionedconditions do hold on these spaces, from which the theorems will follow. Weend this section with a technical statement that will be needed later for thispurpose.

Lemma 2.2. Let M be a smooth manifold and let G be a Lie group actingsmoothly, freely and properly on M . Let h ∈ C∞(M) be constant on each G-orbit, so that it descends to h ∈ C∞(M/G). Then there exist smooth coordinates(x1, . . . , xn, y1, . . . , yk) for M such that (y1, . . . , yk) are coordinates for M/Gand

∂i1+···+ish

∂yi1 · · · ∂yis=

∂i1+···+ish

∂yi1 · · · ∂yisfor any indices i1, . . . , is.

4It is easily seen that with the same arguments one can prove completely analogousconditions in which one approximates ft by (say) the first n eigenspaces instead of just thefirst two.

5The growth property for the eigenvalues is obvious because they are integers; this factwill be remarked later.



Proof. In Theorem 21.10 of [6] one sees that there exist cubic coordinates(x1, . . . , xn, y1, . . . , yk) for M that intersect each G-orbit in either the empty setor a slice (y1, . . . , yk) = (c1, . . . , ck) and such that (y1, . . . , yk) are coordinatesfor M/G. In these coordinates it is clear that

h(x1, . . . , xn, y1, . . . , yk) = h(y1, . . . , yk),

and the statement follows by differentiating both sides. �X

3. Real projective spaces

In this section we demonstrate our result for real projective spaces RPn. Byconstruction, the space of functions on RPn (realized as Sn/G, with G being thetwo element group generated by the antipodal map) is identified (via pullback)with the functions in Sn for which

f(x) = f(−x).

Therefore, one sees that the eigenfunctions for the Laplace-Beltrami operatorin this manifold are precisely the ones on Sn that satisfy f(x) = f(−x). Thatis, the eigenvalues are λj = 2j(2j + n− 1), with the corresponding eigenspacebeing the space of homogeneous harmonic polynomials of degree 2k in n + 1variables [2]. The following proposition is then a detailed analysis of the firstnon-trivial eigenspace, whose ultimate purpose is the verification of condition(C1) of Lemma 2.1.

Proposition 3.1. Let f(x) =∑i,j aijxixj be a real quadratic form in n + 1

variables, with A = (aij) being a symmetric matrix.6 Then f is a minimalMorse function on RPn with distinct critical points if and only if the matrixA = (aij) has distinct eigenvalues.

Proof. We may diagonalize this quadratic form through an orthogonal changeof coordinates y = Bx, and write f(y) = λ1y

21 + · · ·+ λn+1y

2n+1, where λi are

the eigenvalues of A. Write D = diag(λ1, . . . , λn+1). The gradient of f in thesecoordinates (seen as a function in Rn+1) is then ∇f(y) = 2Dy. The proof willbe done once we show that f has exactly n+1 critical points in RPn, all of thembeing non-degenerate, if and only if A has distinct eigenvalues.7 Since RPn isthe quotient of Sn by a properly discontinuous action, it suffices to count thenumber η of critical points for f on the sphere, and then counting the resultingequivalence classes. Since every point on the sphere has orbit two under theantipodal map, the answer will be η/2, thus we need to show that η = 2(n+1).Now, the critical points of f on the sphere will be those y that satisfy

∇f(y) · v = 0 for all v ∈ Ty(Sn) = 〈y〉⊥,6It is easily seen that any quadratic form may be expressed in this symmetric fashion.7Since the Morse-Smale Characteristic of RPn, the minimum number of critical points a

Morse function can have in this space, is n+ 1 [1].



which means2Dy · v = 0 for all v ∈ Ty(Sn) = 〈y〉⊥,

or equivalently 〈y〉⊥ ⊂ 〈Dy〉⊥, and this just says that Dy is a scalar multipleof y. Since D is just the matrix of A in the y coordinates, the desired criticalpoints are precisely the eigenvectors of the transformation A that lie on thesphere.

If A had a repeated eigenvalue, then by intersecting a two-dimensionaleigenspace with the sphere we see that there would be an entire circumfer-ence of critical points, implying that f is not a Morse function on the sphere(since a Morse function has only isolated critical points), so this proves the“only if” part of the statement.

Now suppose the eigenvalues are all distinct, so all eigenspaces are straightlines. Then the desired critical points are precisely the 2(n+ 1) intersections ofthese lines with the sphere (two per line), which in our choice of coordinatesare just the canonical basis vectors {±e1, . . . ,±en+1}, so we get η = 2(n+ 1),as wanted. The critical values are distinct because they are precisely the λi.Now, it remains to see that these critical points ±ei are all non-degenerate. Toprove this, we compute the Hessian in local coordinates and verify that it isinvertible. Fix i, and write f in the local coordinates for Sn around ±ei definedby (u1, . . . , un) = (y1, . . . , yi−1, yi+1, . . . , yn+1). One gets:

f(u1, . . . , un) = λi +∑j 6=i

(λj − λi)u2j

Therefore the gradient is given by

∇f(u) = 2diag(λ1 − λi, . . . , λi−1 − λi, λi+1 − λi, . . . , λn+1 − λi)u

and so the Hessian of f at ei in these coordinates is the matrix

2diag(λ1 − λi, . . . , λi−1 − λi, λi+1 − λi, . . . , λn+1 − λi),

which is invertible because the λi are distinct. �X

This section now ends with the proof of the first of the two main results.

Proof of Theorem 1.4. To prove this theorem we demonstrate that con-ditions (C1) and (C2) of Lemma 2.1 are satisfied. By the remarks at thebeginning of the section, the generic function in the λ1-eigenspace has theform of f in Proposition 3.1, with the additional condition that tr(A) = 0, oran+1,n+1 = −(a11 + · · ·+ ann). Therefore a basis for this eigenspace is

B ={

12xixj

∣∣ 1 ≤ i < j ≤ n+ 1}∪{x2ii − x2

n+1,n+1 | 1 ≤ i < n+ 1}.

The coefficients of f for this basis (of size (n2 + 3n)/2) are the aij for i < jor i = j < n + 1. Diagonalizing A = QTdiag(λ1, . . . , λn+1)Q one sees that



by leaving Q fixed and slightly varying the λi (while keeping the tracelessnesscondition), one can make the eigenvalues distinct, so by continuity (given by thematrix equation) and Proposition 3.1, the set B of condition (C1) in Lemma

2.1 corresponding to our B is dense in R(n2+3n)/2.

To show that B is open, we note first that the coefficients of the character-istic polynomial p of A are continuous functions of the aij (in the usual sense),and the λi depend continuously on the coefficients of p in the following sense:if we consider disjoint ε-neighborhoods of the λi, then the zeroes of pt will, insome order (say β1, . . . , βn+1), satisfy |λi − βi| < ε, whenever pt is obtainedthrough small enough variations of the coefficients of p. This is readily seen tobe a corollary to Rouche’s Theorem,8 and it (together with the fact that A issymmetric which forces all roots to be real) implies that if p has distinct realroots, small variations of the aij will not affect this property, as desired.

Now to check condition (C2), let h2j = πj(f) be the projection of f ontothe jth eigenspace, with h2j being the pulled back homogeneous harmonicpolynomial of degree 2j. By Lemma 2.2, ‖hj‖r = O(1)‖hj‖r. From Section 10in [3], we then obtain

‖h2j‖r = O(1)

(n+ 2j

n

)1/2 (1 + λ

1/2j

)r‖h2j‖L2(Sn).

Since both λj = 2j(2j + n − 1) and(n+2jn

)are polynomials in j (of degrees 2

and n respectively), and ‖h2j‖L2(Sn) ≤ ‖f‖L2(Sn), the desired inequality then

follows immediately, completing the proof. �X

4. Complex projective spaces

This section is dedicated to proving the desired Theorem 1.5 for the complexprojective space CPn. We henceforth identify Cn with R2n as real vector spacesand manifolds by the correspondence

(z1, . . . , zn) = (x1 +√−1y1, . . . , xn +

√−1yn) ≡ (x1, y1, . . . , xn, yn).

The Riemannian manifold CPn with the Fubini-Study metric may then berealized as the quotient S2n+1/U(1), and the (complex-valued) eigenfunctionsfor the Laplacian are the eigenfunctions on the sphere that are invariant undermultiplication by elements of U(1). The most convenient way to characterizethem for the matter at hand is as the bi-homogeneous harmonic polynomialsin (z, z) of bi-degree (k, k) (k = 0, 1, 2, . . .), with corresponding eigenvalues4k(n+k) (cf. [2]). Now, this implies that the real-valued eigenfunctions for thefirst eigenvalue 4(n+1) are precisely the functions f described in the proposition

8Rouche’s Theorem is a standard complex-analytic result that, in rough terms, states thatthe number of zeroes of two analytic functions inside a bounded region is invariant if thesefunctions are close enough in the boundary of the region.



below when A is traceless, since the condition of being real-valued forces thematrix of coefficients to be Hermitian. This result then plays a role analogousto that of Proposition 3.1 in the previous section.

Proposition 4.1. Let f : Cn+1\{0} → R be defined by

f(z) =

∑i,j aijzizj

|z|2,

with A = (aij) being a Hermitian matrix. Then f descends to a minimal Morsefunction with distinct critical values on CPn (realized as the quotient of Cn\{0}obtained by identifying points on the same line) if and only if the matrix A =(aij) has distinct eigenvalues.

Proof. It is clear that f is 0-homogeneous, so we may consider it as a functionon CPn. Similarly to the proof of Proposition 3.1, the fact that A is Hermitianallows us to use the Spectral Theorem to diagonalize A through a unitarychange of coordinates w = Bz. In these coordinates f takes the form

f(w) =

∑n+1i=1 λi|wi|2

|w|2,

where λi are the eigenvalues of A. Replacing B with a permutation of itscolumns allows us to assume λ1 ≤ . . . ≤ λn+1. The sum of the Betti numbers forCPn is n+1, so the minimality condition for f amounts to it having n+1 criticalpoints. The sets Ui = {[w] ∈ CPn | wi 6= 0} together with the inhomogeneouscoordinate functions w = (w1, . . . , wi−1, wi+1, . . . , wn) are a family of smoothcharts covering CPn, hence we may proceed by computing the critical pointsin each Ui through the use of the corresponding local coordinates. In theseinhomogeneous coordinates, f then takes the form

f(w1, . . . , wi−1, wi+1, . . . , wn) =λi +

∑j 6=i λj |wj |2

1 +∑j 6=i |wj |2

=λi +

∑j 6=i λjx

2j + λjy

2j

1 +∑j 6=i x

2j + y2

j

,

This gives

df =−∑j 6=i 2xjdxj + 2yjdyj

(1 +∑j 6=i x

2j + y2

j )2

λi +∑j 6=i

λjx2j + λjy

2j

+

∑j 6=i 2λjxjdxj + λj2yjdyj

1 +∑j 6=i x

2j + y2

j

=

∑j 6=s 2xj

(λj − λi +

∑s6=i(λj − λs)(x2

s + y2s))

dxj

(1 +∑j 6=i x

2j + y2

j )2

+

∑j 6=s 2yj

(λj − λi +

∑s6=i(λj − λs)(x2

s + y2s))

dyj

(1 +∑j 6=i x

2j + y2

j )2.



So df = 0 means that for each j 6= i,

xj

λj − λi +∑s6=i

(λj − λs)(x2s + y2

s)

= 0

and

yj

λj − λi +∑s 6=i

(λj − λs)(x2s + y2

s)

= 0

If λi was a repeated eigenvalue, say λi = λi+1, then letting x1 = . . . = xi−1 =xi+2 = . . . = xn = 0 and y1 = . . . = yi−1 = yi+2 = . . . = yn+1 = 0 it followsthat the equations above hold for any choice of xi+1, yi+1, so the critical pointsof f are not isolated and therefore it is not a Morse function; the “only if” partof the proposition is thus proved.

Conversely, suppose λ1 < . . . < λn+1. Setting j = 1 in the previous twoequations and using the fact that the λs are in increasing order, it follows thatx1 = y1 = 0. Inductively one gets x2, y2, . . . , xi−1, yi−1 = 0. Similarly startingwith j = n+1 and inducting backwards one concludes xj = yj = 0 for each j 6=i. Hence the unique critical point of f in Ui is the point [0, . . . , 1, . . . , 0], wherethe 1 is in the ith position. So there are n+ 1 critical points in total, and theircorresponding values are distinct (they are the λi). To check non-degeneracy,it follows easily from the previous computation that the determinant of theHessian at [0, . . . , 1, . . . , 0] in the ith inhomogeneous coordinates is just

22n(λ1 − λi)2 · · · (λi−1 − λi)2(λi+1 − λi)2 · · · (λn+1 − λi)2,

which is non-zero because the λi are distinct. �X

The previous analysis now allows us to prove the second and final theorem.

Proof of Theorem 1.5. As in the proof of Theorem 1.4, we must verify con-ditions (C1) and (C2) of Lemma 2.1. This time the generic function in theλ1-eigenspace has the form of f in Proposition 4.1, with the condition tr(A) = 0,or an+1,n+1 = −(a11 + · · · + ann). Write aij = bij +

√−1cij . Since aji = aij ,

writing

aijzizj + ajizjzi = bij(zizj + zjzi) +√−1cij(zizj − zjzi),

one sees that the dimension in this case is n(n + 2) and the following set is abasis:

{zizj + zjzi | 1 ≤ i < j ≤ n+ 1} ∪{√−1(zizj − zjzi) | 1 ≤ i < j ≤ n+ 1

}∪{z2ii − z2

n+1,n+1 | 1 ≤ i < n+ 1}.



The coefficients of f for this basis are the bij and the cij for i < j, and the biifor i < n+1 (since cii = 0). Noticing that the bij , cij and the aij are bijectivelyand bicontinuously related, we may diagonalize A = Q∗diag(λ1, . . . , λn+1)Qand finish verifying the same condition (C1) with the same argument as thatof Theorem 1.4. Now, for condition (C2), the argument in Theorem 1.4 (againappealing to Lemma 2.1) gives the estimate

‖h2j‖r = O(1)

(2n+ 1 + 2j

2n+ 1

)1/2 (1 + λ

1/2j

)r‖h2j‖L2(S2n+1)

for the projection of f onto the jth eigenspace, and the result follows in thesame way since λj = 4j(n+j) and

(2n+1+2j

2n+1

)are polynomials (of degrees 2 and

2n+ 1) in j. �X

Acknowledgements. We thank professor Gonzalo Garcıa Camacho for hisunconditional support and valuable advice during the development of this work.We also thank the anonymous referee for reading and evaluating our work, aswell as writing a report containing interesting and valuable remarks.

References

[1] D. Andrica, D. Magra, and C. Pintea, The circular Morse-Smale character-istic of closed surfaces, Bull. math. Soc. Sci. Math. Roum. 57 (2014), no. 3,235–242.

[2] M. Berger, P. Gauduchon, and E. Mazet, Le Spectre d’une Variete Rieman-nienne, Springer-Verlag, 1971.

[3] P. Garret, Harmonic analysis on spheres, II, http://www-users.math.

umn.edu/~garrett/m/mfms/notes_c/spheres_II.pdf.

[4] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities,Springer-Verlag, 1973.

[5] M. W. Hirsch, Differential Topology, Springer-Verlag, 1976.

[6] J. Lee, Introduction to Smooth Manifolds, Springer, 2013.

[7] C. Cadavid Moreno and J. D. Velez, A remark on the Heat Equationand minimal Morse functions on Tori and Spheres, Ingenierıa y Ciencia9 (2013), no. 17, 11–20.

(Recibido en septiembre de 2016. Aceptado en marzo de 2017)



Departamento de Matematicas

Facultad de Ciencias

Universidad del Valle

Calle 13 # 100-00

Cali, Colombia

e-mail: [email protected]

Escuela de Matematicas

Facultad de Ciencias

Universidad Nacional de Colombia Sede Medellın

Calle 59a # 63-20

Medellin, Colombia

e-mail: [email protected]


Documents

Heat equation and stable minimal Morse functions … · Heat equation and stable minimal Morse functions on real ... Operador de Laplace-Beltrami, ... proof of the analogous result